Topic
Velocity gradient
About: Velocity gradient is a research topic. Over the lifetime, 3013 publications have been published within this topic receiving 77120 citations.
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TL;DR: In this paper, a simple and economical iterative scheme is presented for calculating turbulent shear layers with significant pressure gradients normal to the plane of the layer, such as occur on highly-curved surfaces or near the trailing edges of lifting airfoils, and for matching the shear-layer calculations to calculations of the inviscid external flow.
Abstract: A simple and economical iterative scheme is presented for calculating turbulent shear layers with significant pressure gradients normal to the plane of the layer, such as occur on highly-curved surfaces or near the trailing edges of lifting airfoils, and for matching the shear-layer calculations to calculations of the inviscid external flow. The iteration required to solve the elliptic equations describing the shear layer is combined with the iteration needed for the matching, and the finite-difference solution of the normal-component momentum equation is a simple quadrature at each iteration. Therefore computing time is little greater than in conventional displacement-surface calculations that ignore normal pressure gradients. OUNDARY-layer (thin-shear-layer) equations are derived from the Navier-Stokes equations by assuming that streamwise (x) gradients of velocity are small compared to velocity gradients normal to the surface (y). The resulting simplifications1 include the disappearance of the pressure gradient normal to the surface, from which follows the smallness of the velocity gradient dU/dy in the "inviscid" flow just outside the shear layer. This in turn leads to the concept of a displacement thickness to represent the displacement, nominally independent of y, of the "inviscid" flow streamlines near the shear layer. It is important to note that once the basic assumption fails, all the simplifications disappear. If the shear layer changes rapidly in the x direction, so that dV/dx is large, normal pressure gradients are im- portant both within the layer and outside it; not only does the displacement thickness fail to represent the displacement of the external flow, but its definition, even as a mere shear-layer parameter, becomes ambiguous. It will be seen that correc- tions based on the surface radius of curvature may be highly inaccurate. All but the most rapidly changing viscous or turbulent flows will still be recognizable as fairly thin shear layers, and the terms in the Navier-Stokes equations that are neglected in the boundary-layer equations will still be v fairly small. Therefore, thin-shear-layer concepts are still useful in analytic and computational work. In particular, the effects of the normal pressure gradient on the shear layer will be small enough to be included by iterative improvement of a con- ventional marching calculation rather than by a fully elliptic calculation, and this more economical approach has been adopted in the present work.
25 citations
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TL;DR: In this paper, the second-order macroscopic constitutive equation was derived from the kinetic Boltzmann equation via the balanced closure and cumulant expansion, and new analytical solutions to the Knudsen layer in Couette flow, in conjunction with algebraic nonlinearly coupled secondorder constitutive and Maxwell velocity slip and Smoluchowski temperature jump models, were derived.
Abstract: The Knudsen layer, found in the region of gas flow very close (in order of a few mean free paths) to the solid surfaces, plays a critical role in accurately modeling rarefied and micro-scale gases. In various previous investigations, abnormal behaviors at high Knudsen numbers such as nonlinear velocity profile, velocity gradient singularity, and pronounced thermal effect are identified to exist in the Knudsen layer. However, some behaviors, in particular, the velocity gradient singularity near the surface and higher temperature, remain elusive in the continuum framework. In this study, based on the second-order macroscopic constitutive equation recently derived from the kinetic Boltzmann equation via the balanced closure and cumulant expansion [R. S. Myong, “On the high Mach number shock structure singularity caused by overreach of Maxwellian molecules,” Phys. Fluids 26(5), 056102 (2014)], the macroscopic second-order constitutive and slip-jump models that are able to explain qualitatively all the known non-classical and non-isothermal behaviors are proposed. As a result, new analytical solutions to the Knudsen layer in Couette flow, in conjunction with the algebraic nonlinearly coupled second-order constitutive and Maxwell velocity slip and Smoluchowski temperature jump models, are derived. It was shown that the velocity gradient singularity in the Knudsen layer can be explained within the continuum framework, when the nonlinearity of the constitutive model is morphed into the determination of the velocity slip in the nonlinear slip and jump model. Also, the smaller velocity slip and shear stress are shown to be caused by the shear-thinning property of the second-order constitutive model, that is, vanishing effective viscosity at high Knudsen number.
25 citations
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TL;DR: In this paper, the spatial orientation of rigid ellipsoidal particles was analyzed as proceeding in a dilute solution flowing in a velocity field with parallel gradient, i.e., in a field characterized by the deformation rate tensor.
Abstract: The spatial orientation of rigid ellipsoidal particles was analyzed as proceeding in a dilute solution flowing in a velocity field with parallel gradient, i.e., in a field characterized by the deformation rate tensor:
On the basis of general relations given by Jeffery, the hydrodynamic equations of motion of a single ellipsoid were obtained as Ψ = 0, φ = 0, θ = −¾qR sin 2θ, where q = ∂Vκ/∂κ is the parallel velocity gradient and R = (a2 − b2)/(a2 + b2) is the shape coefficient of ellipsoids. Considering the action of velocity field and that of Brownian motion (rotational diffusion), a distribution density function ρ(t, θ) was derived, which describes the spatial orientation of the axes of symmetry of the ellipsoids:
where
is the steady-state distribution. In a similar way, the axial orientation factor f0 = 1 − 3/2 sin2θ was obtained:
25 citations
01 Jan 2002
TL;DR: In this article, a weakly inhomogeneous and unsteady form of the rapid distortion theory (RDT) was used to study the growth of small temporal and spatial perturbations in the large-scale mean stratification and mean velocity profile in a freely decaying, stably stratified homogeneous turbulent flow with r.m.s.
Abstract: The initial evolution of the momentum and buoyancy fluxes in a freely decaying, stably stratified homogeneous turbulent flow with r.m.s. velocity u′0 and integral lengthscale l0 is calculated using a weakly inhomogeneous and unsteady form of the rapid distortion theory (RDT) in order to study the growth of small temporal and spatial perturbations in the large-scale mean stratification N(z, t) and mean velocity profile u(z, t) (here N is the local Brunt–Vaisala frequency and u is the local velocity of the horizontal mean flow) when the ratio of buoyancy forces to inertial forces is large, i.e. Nl0/u′0[dbl greater-than sign]1. The lengthscale L of the perturbations in the mean profiles of stratification and shear is assumed to be large compared to l0 and the presence of a uniform background mean shear can be taken into account in the model provided that the inertial shear forces are still weaker than the buoyancy forces, i.e. when the Richardson number Ri = (N/[partial partial differential]zu)2[dbl greater-than sign]1 at each height. When a mean shear perturbation is introduced initially with no uniform background mean shear and uniform stratification, the analysis shows that the perturbations in the mean flow profile grow on a timescale of order N-1. When the mean density profile is perturbed initially in the absence of a background mean shear, layers with significant density gradient fluctuations grow on a timescale of order N−10 (where N0 is the order of magnitude of the initial Brunt–Vaisala frequency) without any associated mean velocity gradients in the layers. These results are in good agreement with the direct numerical simulations performed by Galmiche et al. (2002) and are consistent with the earlier physically based conjectures made by Phillips (1972) and Posmentier (1977). The model also shows that when there is a background mean shear in combination with perturbations in the mean stratification, negative shear stresses develop which cause the mean velocity gradient to grow in the density layers. The linear analysis for short times indicates that the scale on which the mean perturbations grow fastest is of order u′0/N0, which is consistent with the experiments of Park et al. (1994). We conclude that linear mechanisms are widely involved in the formation of shear and density layers in stratified flows as is observed in some laboratory experiments and geophysical flows, but note that the layers are also significantly influenced by nonlinear and dissipative processes at large times.
25 citations
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TL;DR: In this paper, the authors show that short-wavelength random velocity fluctuations of only 0.5-1% superimposed on negative velocity gradients are sufficient for generating Pn phases and this implies that an observed Pn wave does not necessitate a positive upper mantle velocity gradient.
Abstract: Pn phases are observed along many refraction seismic profiles and are common in earthquake records. Their velocities usually range from 7.8 to 8.2 km s−1. Classical ray theory used to interpret these observations implies a positive upper mantle velocity gradient. However, a wide spread positive velocity gradient in the lithospheric mantle is not expected from petrological and petrophysical data. Laboratory velocity measurements at elevated temparatures and pressures suggest positive velocity gradients only for very low heat flow values (≤40 mW m−2). Higher heat flow causes negative gradients. Consequently, petrological models of the upper mantle would restrict Pn observations to Precambrian shields and old platforms, contrary to observations. We overcome this contradiction by considering media that contain random velocity fluctuations superimposed on positive or negative velocity gradients. In both cases, these structures generate Pn phases by wide-angle scattered waves. Short-wavelength random velocity fluctuations of only 0.5–1% superimposed on negative velocity gradients are sufficient for generating Pn phases. Consequently, this implies that an observed Pn wave does not necessitate a positive upper mantle velocity gradient. For a peridotitic upper mantle, fluctuations of this size can be explained by slightly varying the relative proportions of its mineralogical constituents. Anisotropy is likely to contribute to the inferred fluctuations.
25 citations